 
Summary: HOMOMORPHISMS BETWEEN MAPPING CLASS
GROUPS
JAVIER ARAMAYONA & JUAN SOUTO
Abstract. Suppose that X and Y are surfaces of finite topologi
cal type, where X has genus g 6 and Y has genus at most 2g1;
in addition, suppose that Y is not closed if it has genus 2g  1.
Our main result asserts that every nontrivial homomorphism
Map(X) Map(Y ) is induced by an embedding, i.e. a combina
tion of forgetting punctures, deleting boundary components and
subsurface embeddings. In particular, if X has no boundary then
every nontrivial endomorphism Map(X) Map(X) is in fact an
isomorphism.
As an application of our main theorem we obtain that, under
the same hypotheses on genus, if X and Y have finite analytic
type then every nonconstant holomorphic map M(X) M(Y )
between the corresponding moduli spaces is a forgetful map. In
particular, there are no such holomorphic maps unless X and Y
have the same genus and Y has at most as many marked points as
X.
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